Parametric Bilinear Generalized Approximate Message Passing

TL;DR

This paper introduces a generalized approximate message passing (G-AMP) scheme for estimating parameters in bilinear models from noisy measurements, applicable to various problems like self-calibration and blind deconvolution.

Contribution

It extends bilinear G-AMP to handle arbitrary likelihood functions and tensor structures, broadening its applicability to complex bilinear estimation problems.

Findings

The proposed method accurately estimates parameters in bilinear models.

It demonstrates computational efficiency in large-system limits.

Numerical experiments validate the approach's effectiveness.

Abstract

We propose a scheme to estimate the parameters $b_i$ and $c_j$ of the bilinear form $z_m=\sum_{i,j} b_i z_m^{(i,j)} c_j$ from noisy measurements $\{y_m\}_{m=1}^M$, where $y_m$ and $z_m$ are related through an arbitrary likelihood function and $z_m^{(i,j)}$ are known. Our scheme is based on generalized approximate message passing (G-AMP): it treats $b_i$ and $c_j$ as random variables and $z_m^{(i,j)}$ as an i.i.d.\ Gaussian 3-way tensor in order to derive a tractable simplification of the sum-product algorithm in the large-system limit. It generalizes previous instances of bilinear G-AMP, such as those that estimate matrices $\boldsymbol{B}$ and $\boldsymbol{C}$ from a noisy measurement of $\boldsymbol{Z}=\boldsymbol{BC}$, allowing the application of AMP methods to problems such as self-calibration, blind deconvolution, and matrix compressive sensing. Numerical experiments confirm the accuracy and computational efficiency of the proposed approach.